Learn about a power series, a series of terms in the form of (a_0 + a_1x + a_2x^2 + cdots), used to represent functions and solve equations in mathematics.

A power series is a series that represents a function as an infinite sum of terms, each of which is a polynomial multiplied by a coefficient. The general form of a power series is:

$$sum_{n=0}^{infty} a_n cdot (x - c)^n = a_0 + a_1 cdot (x - c) + a_2 cdot (x - c)^2 + a_3 cdot (x - c)^3 + ldots$$

**Terms:** Each term consists of a coefficient, denoted by (a_n), multiplied by a variable raised to the (n^{th}) power.

**Variable:** The variable (x) represents the input to the function. The point (c) is the center of the power series and indicates where the series is expanded around.

**Coefficients:** The coefficients (a_n) determine the shape of the function represented by the power series. The behavior of the function is determined by how the coefficients change as (n) varies.

Power series are widely used in mathematics, physics, engineering, and other fields. They provide a way to represent complicated functions in a simpler form that is easier to analyze and manipulate. They are particularly useful in calculus for approximating functions, solving differential equations, and studying the behavior of functions near a specific point.

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